Lotka Volterra absolute fishing problem - Revision history

ClemensZeile: /* Mathematical formulation */

2019-10-14T11:36:33Z

Mathematical formulation

← Older revision		Revision as of 11:36, 14 October 2019
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	</p>		</p>

	Here the differential states <math>(x_0, x_1)</math> describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation <math>\min \; x_2(t_f)</math>. This problem variant allows to choose between ~~three~~ different fishing options.		Here the differential states <math>(x_0, x_1)</math> describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation <math>\min \; x_2(t_f)</math>. This problem variant allows to choose between five different fishing options.

	== Parameters ==		== Parameters ==

ClemensZeile: /* Reference Solutions */

2019-10-14T11:10:19Z

Reference Solutions

← Older revision		Revision as of 11:10, 14 October 2019
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
	Image:~~MmlotkaRelaxed_12000_30_1~~.~~png~~\| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=12000, \, n_u=~~400~~</math>.		Image:Lotka_abs_fish_relaxed_12000_80.pdf\| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=12000, \, n_u=150</math>.
	Image:~~MmlotkaCIA~~ 12000 ~~30 1~~.~~png~~\| ~~Optimal binary controls and states~~ determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=~~400~~</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.		Image:Lotka_abs_fish_CIA_states_12000_80.pdf\| Differential states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=150</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
			Image:Lotka_abs_fish_CIA_controls_12000_80.pdf\| Binary control determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=150</math>. The relaxed controls were approximated by Combinatorial Integral Approximation. The fishing control shows a lot of chattering.
	</gallery>		</gallery>

ClemensZeile at 10:56, 14 October 2019

2019-10-14T10:56:25Z

← Older revision		Revision as of 10:56, 14 October 2019
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	== Reference Solutions ==		== Reference Solutions ==

	If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> ~~instead of the~~ binary ~~choice <math>\{0,1\}</math>~~, the optimal solution can be determined by means of direct optimal control.		If the problem is relaxed, i.e., we demand that <math>w(t)</math> is in the continuous interval <math>[0, 1]</math> rather than being binary, the optimal solution can be determined by means of direct optimal control.

	The optimal objective value of the relaxed problem with <math> n_t=12000, \, n_u=~~400~~ </math> is <math>x_2(t_f) =1.~~82875272~~</math>. The objective value of the binary controls obtained by Combinatorial Integral ~~Approimation~~ (CIA) is <math>x_2(t_f) =1.~~82878681~~</math>.		The optimal objective value of the relaxed problem with <math> n_t=12000, \, n_u=150 </math> is <math>x_2(t_f) =0.345768563</math>. The objective value of the solution with binary controls obtained by Combinatorial Integral Approximation (CIA) is <math>x_2(t_f) =0.348617982</math>.

	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">

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2019-10-14T07:24:38Z

Created page with "{{Dimensions |nd = 1 |nx = 3 |nw = 5 |nre = 3 }}This site describ..."

New page

{{Dimensions
|nd = 1
|nx = 3
|nw = 5
|nre = 3
}}This site describes a Lotka Volterra variant with five binary controls that all represent fishing of an absolute biomass.

== Mathematical formulation ==

The mixed-integer optimal control problem is given by

<p>
<math>
\begin{array}{llclr}
\displaystyle \min_{x, w} & x_2(t_f) \\[1.5ex]
\mbox{s.t.}
& \dot{x}_0 & = & x_0 - x_0 x_1 - \; \sum\limits_{i=1}^{5} c_{0,i}\; w_i, \\
& \dot{x}_1 & = & - x_1 + x_0 x_1 - \; \sum\limits_{i=1}^{5} c_{1,i}\; w_i, \\
& \dot{x}_2 & = & (x_0 - 1)^2 + (x_1 - 1)^2, \\[1.5ex]
& x(0) &=& (0.5, 0.7, 0)^T, \\
& \sum\limits_{i=1}^{5}w_i(t) &=& 1, \\
& w_i(t) &\in& \{0, 1\}, \quad i=1\ldots 5.
\end{array}
</math>
</p>

Here the differential states <math>(x_0, x_1)</math> describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation <math>\min \; x_2(t_f)</math>. This problem variant allows to choose between three different fishing options.

== Parameters ==

These fixed values are used within the model.

<math>
\begin{array}{rcl}
[t_0, t_f] &=& [0, 12],\\
(c_{0,1}, c_{1,1}) &=& (0.2, 0.1),\\
(c_{0,2}, c_{1,2}) &=& (0.4, 0.2),\\
(c_{0,3}, c_{1,3}) &=& (0.01, 0.1),\\
(c_{0,4}, c_{1,4}) &=& (0, 0),\\
(c_{0,5}, c_{1,5}) &=& (-0.1, -0.2).
\end{array}
</math>

== Reference Solutions ==

If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of direct optimal control.

The optimal objective value of the relaxed problem with <math> n_t=12000, \, n_u=400 </math> is <math>x_2(t_f) =1.82875272</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>x_2(t_f) =1.82878681</math>.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:MmlotkaRelaxed_12000_30_1.png| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=12000, \, n_u=400</math>.
Image:MmlotkaCIA 12000 30 1.png| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=400</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
</gallery>


[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Tracking objective]]
[[Category:Chattering]]
[[Category:Sensitivity-seeking arcs]]
[[Category:Population dynamics]]